MATH 134  Math and Politics
NIM
•
Coins are arranged in columns; two players alternate turns; the
player who takes the last coin wins.
•
Game tree organized way to look at all possible configurations.
o
Nodes configurations.
o
Edges possible moves.
o
Branch path from start to leaf (possible game).
•
To determine who has a winning strategy, use Backwards
Induction/Pruning/Dynamic Programming
.
o
Look at nodes that have only leaves below it.
•
We assume that at every stage, the player will act rationally, in his
best interest.
Dollar Auction
•
Invented by Shubik in 1971 as a model for escalation (ex: Cold
War) and solved by O’Neil in 1986 (for conservative convention.
•
2 players bid for same prize; on 1
st
turn player 1 must bid, after
players can bid or pass (game ends); both players’ bid goes to
auctioneer; s=stakes; b=bankroll.
•
Goal: to make biggest profit or smallest loss.
•
Punishing convention
 pick worst outcome for opponent (real life
ex: poker).
•
Conservative convention
 minimize risk by picking lowest bid
(real life ex: regular auction).
•
O’Neil’s Theorem
 if bankroll is b>0$ and stakes are s>1 and the
conservative convention is followed, then there is an optimal 1
st
bid for P1 such that after that bid, P2 will pass.
It can be determined by: take bankroll b and repeatedly subtract s1
from b. The last value obtained before reaching zero or a negative
# is the optimal bid.
•
For problems, would need a bid #, stake #, and would have to show
a game tree showing all possible results, then need to label win
loss total with (+/x,+/y). After, need to show either punishing or
conservative convention, as well as backwards induction.
2x2 Ordinal Games
•
2 players, row and column, & each has two alternatives to choose
between (cooperate [C] and noncooperate [N]); play consists of
N; each player ranks outcome (out of 4 possible) according to
preference, 4 being best & 1 being worst (rankingordinal).
•
A game in which row and column prefer to cooperate is called a
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 Fall '07
 RASSART
 Math, The Land

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